\u4eba\u5de5\u77e5\u80fd(AI)\u3084\u6a5f\u68b0\u5b66\u7fd2\u3068\u3044\u3048\u3070\u8d85\u7403\u304c\u767b\u5834\u3059\u308b\u3053\u3068\u304c\u591a\u3044\u3067\u3059\u304c(\u203b\u500b\u4eba\u306e\u611f\u60f3\u3067\u3059\u3002)\u3001\u8d85\u7403\u306e\u4f53\u7a4d\u3068\u3044\u3048\u3070\u30e4\u30b3\u30d3\u30a2\u30f3\u3068\u3044\u3046\u3053\u3068\u3067\u3001$n$\u6b21\u5143\u306e\u76f4\u4ea4\u5ea7\u6a19\u304b\u3089\u7403\u9762\u5ea7\u6a19\u3078\u306e\u5909\u6570\u5909\u63db\u306e\u305f\u3081\u306e\u30e4\u30b3\u30d3\u884c\u5217(\u30e4\u30b3\u30d3\u30a2\u30f3)\u306b\u3064\u3044\u3066\u66f8\u304d\u307e\u3059\u3002<\/p>\n
$n$\u6b21\u5143\u7a7a\u9593\u306e\u70b9$\\boldsymbol{x}$\u306e\u4f4d\u7f6e\u3092$n$\u6b21\u5143\u306e\u76f4\u4ea4\u5ea7\u6a19\u7cfb\u306e\u5ea7\u6a19\u5024\u3067\u8868\u3059\u3068\u2026
\n\\begin{align}
\n\\boldsymbol{x} &= (x_1, x_2, \\cdots, x_n)^{T} \\label{eq:euclid}
\n\\end{align}
\n$\\boldsymbol{x} \\in \\mathbb{R}^n$\u3068\u7f6e\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n
\u4e00\u65b9\u3067\u3001\u540c\u3058\u70b9\u3092$n$\u6b21\u5143\u306e\u7403\u9762\u5ea7\u6a19\u7cfb\u306e\u5ea7\u6a19\u5024\u3067\u8868\u3059\u3053\u3068\u3082\u3067\u304d\u3066\u3001
\n\\begin{align}
\n\\boldsymbol{x} &= (r, \\phi_1, \\cdots, \\phi_{n-1})^{T} \\nonumber \\cr
\n&{} \\left(r \\ge 0, 0 \\le \\phi_k \\le \\pi \\, (1 \\le k \\le n-2), 0 \\le \\phi_{n-1} \\lt 2\\pi \\right) \\label{eq:spherical_coordinate}
\n\\end{align}
\n\u3068\u7f6e\u304f\u3053\u3068\u3082\u3067\u304d\u307e\u3059\u3002<\/p>\n
(\\ref{eq:euclid})\u5f0f\u306e$x_1, x_2, \\cdots, x_n$\u3068(\\ref{eq:spherical_coordinate})\u5f0f\u306e$r, \\phi_1, \\cdots, \\phi_{n-1}$\u306e\u9593\u306b\u306f\u4ee5\u4e0b\u306e\u95a2\u4fc2\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002
\n\\begin{align}
\n\\begin{pmatrix}
\nx_1 \\cr
\nx_2 \\cr
\n\\vdots \\cr
\nx_{n-1} \\cr
\nx_n
\n\\end{pmatrix}&= \\left( \\begin{array}{l}
\nr\\sin\\phi_1\\sin\\phi_2 \\cdots \\sin\\phi_{n-2}\\sin\\phi_{n-1} \\cr
\nr\\sin\\phi_1\\sin\\phi_2 \\cdots \\sin\\phi_{n-2}\\cos\\phi_{n-1} \\cr
\n\\qquad\\qquad\\vdots \\cr
\nr\\sin\\phi_1\\cos\\phi_2 \\cr
\nr\\cos\\phi_1
\n\\end{array} \\right) \\label{eq:euclid_spherical}
\n\\end{align}<\/p>\n
\u306a\u304a\u3001(\\ref{eq:euclid_spherical})\u5f0f\u306e\u53f3\u8fba\u306f\u3001
\n\\begin{align}
\nx_k &= \\begin{cases}
\nr \\displaystyle\\prod_{i=1}^{n-1}\\sin\\phi_i & (k = 1)\\cr
\nr \\cos\\phi_{n-k+1}\\displaystyle\\prod_{i=1}^{n-k}\\sin\\phi_i & (k \\gt 1) \\cr
\n\\end{cases}\\label{eq:x_n}
\n\\end{align}
\n\u3068\u307e\u3068\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n
$\\boldsymbol{x}$\u3092$r, \\phi_1, \\cdots, \\phi_{n-1}$\u306e\u95a2\u6570\u3068\u8003\u3048\u308b\u3068\u3001(\\ref{eq:euclid_spherical})\u5f0f\u306e\u95a2\u4fc2\u3088\u308a\u30e4\u30b3\u30d3\u884c\u5217\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002
\n\\begin{align}
\n\\dfrac{\\partial(x_1, x_2, \\cdots, x_n)}{\\partial(r, \\phi_1, \\cdots, \\phi_{n-1})} &= \\begin{pmatrix}
\n\\dfrac{\\partial x_1}{\\partial r} & \\dfrac{\\partial x_1}{\\partial \\phi_1} & \\cdots & \\displaystyle\\frac{\\partial x_1}{\\partial \\phi_{n-1}} \\cr
\n\\vdots & \\vdots & \\ddots & \\vdots \\cr
\n\\dfrac{\\partial x_n}{\\partial r} & \\dfrac{\\partial x_n}{\\partial \\phi_1} & \\cdots & \\displaystyle\\frac{\\partial x_n}{\\partial \\phi_{n-1}}
\n\\end{pmatrix} \\label{eq:jacobian_first}
\n\\end{align}
\n(\\ref{eq:jacobian_first})\u5f0f\u306e\u53f3\u8fba\u306e\u5404\u6210\u5206\u306f(\\ref{eq:x_n})\u5f0f\u3092\u7528\u3044\u308b\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002
\n\\begin{align}
\n\\dfrac{\\partial x_k}{\\partial r} &= \\begin{cases}
\n\\displaystyle\\prod_{i=1}^{n-1}\\sin\\phi_i & (k = 1)\\cr
\n\\cos\\phi_{n-k+1}\\displaystyle\\prod_{i=1}^{n-k}\\sin\\phi_i & (k \\gt 1) \\cr
\n\\end{cases} \\label{eq:diffxkr} \\cr
\n\\dfrac{\\partial x_1}{\\partial \\phi_l} &= r \\cos\\phi_l\\displaystyle\\prod_{i=1,i \\ne l}^{n-1}\\sin\\phi_i \\label{eq:diffxonel} \\cr
\n\\dfrac{\\partial x_k}{\\partial \\phi_l} &= \\begin{cases}
\nr \\cos\\phi_{n-k+1}\\cos\\phi_l\\displaystyle\\prod_{i=1,i \\ne l}^{n-k}\\sin\\phi_i & (x \\gt 1, l \\lt n-k+1) \\cr
\n-r \\displaystyle\\prod_{i=1}^{n-k+1}\\sin\\phi_i & (x \\gt 1, l = n-k+1) \\cr
\n0 & (x \\gt 1, l \\gt n-k+1) \\label{eq:diffxkl}
\n\\end{cases}
\n\\end{align}
\n(\\ref{eq:diffxkl})\u5f0f\u3088\u308a\u3001(\\ref{eq:jacobian_first})\u5f0f\u306e\u53f3\u4e0a\u304b\u3089\u5de6\u4e0b\u306e\u5bfe\u89d2\u7dda\u306e\u3061\u3087\u3063\u3068\u4e0b\u306e\u6210\u5206\u304b\u3089\u3055\u3089\u306b\u4e0b\u5074\u306e\u307b\u307c\u534a\u5206\u306e\u500b\u6570(\u53b3\u5bc6\u306b\u306f\u534a\u5206\u3088\u308a\u3082\u5c11\u306a\u3044\u500b\u6570\u3067\u3059\u304c)\u306e\u6210\u5206\u304c0\u306b\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n
\u7279\u306b\u3001$n$\u5217\u306b\u3064\u3044\u3066\u306f$(3,n)$\u6210\u5206\u304b\u3089$(n,n)$\u6210\u5206\u307e\u3067\u306f0\u306b\u306a\u308b\u3053\u3068\u3001\u306a\u3089\u3073\u306b$\\phi_{n-1}$\u304c\u73fe\u308c\u308b\u6210\u5206\u306f\u7b2c1\u884c\u53ca\u3073\u7b2c2\u884c\u306e\u6210\u5206\u306b\u9650\u5b9a\u3055\u308c\u308b\u3053\u3068\u304b\u3089\u3001(\\ref{eq:jacobian_first})\u5f0f\u306e\u53f3\u8fba\u306f(\\ref{eq:diffxkr}),(\\ref{eq:diffxonel})\u5f0f\u53ca\u3073(\\ref{eq:diffxkl})\u5f0f\u306e\u7d50\u679c\u3088\u308a\u3001(\\ref{eq:jacobian_second})\u5f0f\u306e\u5f62\u5f0f\u306b\u306a\u3063\u3066\u3044\u308b\u3068\u8003\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002
\n\\begin{align}
\n\\dfrac{\\partial(x_1, x_2, \\cdots, x_n)}{\\partial(r, \\phi_1, \\cdots, \\phi_{n-1})} &= \\left(
\n\\begin{array}{ccc|c}
\n\\displaystyle\\prod_{i=1}^{n-1}\\sin\\phi_i & r \\cos\\phi_1\\displaystyle\\prod_{i=2}^{n-1}\\sin\\phi_i & \\cdots & r \\cos\\phi_{n-1}\\displaystyle\\prod_{i=1}^{n-2}\\sin\\phi_i \\cr
\n\\cos\\phi_{n-1}\\displaystyle\\prod_{i=1}^{n-2}\\sin\\phi_i & r \\cos\\phi_{n-1}\\cos\\phi_1\\displaystyle\\prod_{i=2}^{n-2}\\sin\\phi_i & \\cdots & -r \\displaystyle\\prod_{i=1}^{n-k+1}\\sin\\phi_i \\cr \\hline
\n\\vdots & \\vdots & \\ddots & {\\bf 0} \\cr
\n\\hline
\n\\cos\\phi_1 & -r \\sin\\phi_1 & \\cdots & 0
\n\\end{array}
\n\\right) \\label{eq:jacobian_second}
\n\\end{align}<\/p>\n
\u30e4\u30b3\u30d3\u884c\u5217\u3092\u8a08\u7b97\u3059\u308b\u3068\u6b21\u306f\u5909\u6570\u5909\u63db\u306b\u5fc5\u8981\u306a\u884c\u5217\u5f0f(\u30e4\u30b3\u30d3\u884c\u5217\u5f0f)\u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u304c\u4e00\u822c\u7684\u3067\u3059\u304c\u3001\u9577\u304f\u306a\u308a\u305d\u3046\u306a\u306e\u3067\u3053\u3053\u3067\u3044\u3063\u305f\u3093\u3072\u3068\u533a\u5207\u308a\u3068\u3057\u3001\u6b21\u306e\u8a18\u4e8b\u3042\u305f\u308a\u3067\u7d9a\u304d\u3092\u66f8\u304d\u307e\u3059\u3002<\/p>\n
\u3053\u306e\u8a18\u4e8b\u306f\u4ee5\u4e0a\u3067\u3059\u304c\u3001\u81ea\u5206\u3082\u3088\u304f\u53c2\u7167\u3057\u3066\u3044\u308b\u672c\u3092\u7d39\u4ecb\u3057\u307e\u3059\u306e\u3067\u3001\u3088\u308d\u3057\u304b\u3063\u305f\u3089\u3054\u89a7\u3044\u305f\u3060\u3051\u308b\u3068\u5e78\u3044\u3067\u3059\u3002<\/p>\n